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128                     ANALYTICAL MECHANICS
the variation of the gravitational attraction must be taken into account. According to the law of gravitational attraction the force between two gravitating spherical bodies is of the following form:
where m and m' are the masses of the spheres, r is the distance between their centers, and 7 is a positive constant. The negative sign indicates the fact that r is measured in a direction opposed to that in which F acts. When the gravitating bodies are the earth and a body which is small com-
pared with the earth 7 = ~~: , where M is the mass of the
earth, a its radius, and g the gravitational acceleration on the surface of the earth. In order to show this observe that when the body is on the surface of the earth, that is, when r = a, the force is -mg, the weight of the body. Therefore replacing in equation (1) F by-mg and mf by M and solving for 7 we obtain
Substituting in equation (1) this value of 7 we get
for the force which acts upon a body of mass m during its. fall towards the earth. Therefore the force equation is
Dropping m from both sides of equation (4) and writing v ~
for -j- we obtain at